How to use Blender object rotation style using Maya Navigation?
I don't know how it work in Maya, but in Blender you have this options:

1: Turn ON "Transformation manipulate"  

2: Use this shortcuts: (I recommend)


"R" and after "X" = rotation arround X axis in global coordinates
"R" and after two times "X" = rotation around X axis - in local coordinates
"R" and after "SHIFT+X" - for rotate around Y and Z axes (no X)


This work for all axes obliviously and for "G" - grab, "S" - scale and "R" - rotation.